*57 
by a Method of Differences. 
tive ; in order to get the sum by Theorem I. But the sum of 
a series whose terms are alternately positive and negative, 
may be obtained from the sum of a similar series, whose 
terms are all positive by a mere substitution ; thus if the 
sum of the series, a sine of rz — b sine of r -f 5- . % -f c sine of 
r-j-25 . % — &c. were required, put r%=i8o° — pz, and 52=180 
— qz, therefore the sine of rz= sine of 180° — ^>2=sine of pz, 
sine of r-j-5.2=sine of 3 6 o°—p-\-q.z= — sine of p-\-q.z, sine 
of r+25 . 2=sine 540 0 — p-{- zq . «=sine of p-\-v.q .z, See.; and 
consequently the sum of the series, a . sine of rz — b . sine of 
r-\-s.z-\-c. sine of r-\- c 2s.z — Sec— the sum of the series, a sine 
of pz-^-b sine of p-\-q . z-\-c sine of p-\-zq . z &c. ; and by the 
like substitution may the sum of a series of cosines, whose 
terms are alternately positive and negative, be deduced from 
the sum of a series of cosines, whose terms are all positive : 
all this requires the functional values of p and q to be distinct* 
otherwise the substitution cannot be effected ; but the said 
sum may be deduced at once by the following 
Theorem IL 
If there be a series, a. sine of pz—b . sine of p-\-q .z-{-c . sine 
ofp-\- %q . z—d, sine of^-f“3^ • % &c.=s, then shall 
a . sine of p'z — a'. sineof^-J-q.^-j-^- s i ne of p'-\-2q.z Scc.—s f 
a sine of p"z — a " sine of p" -\-q.z-\-b" sine of p"-\-*iq.z Sec.—s 11 
a sine of p m z — a sine of p"'-\-q.z-\-b" sine of ^+2 ~q.z Sec.— s'" 
&c. Sec . Sec. Sec. 
